Polynomial Division Questions

As we know, the polynomial division is one of the important concepts of Class 10 maths. Polynomial division questions and answers are given here to help students learn the division of polynomials by a monomial, binomial and another polynomial. In this article, you will get solved questions on polynomial division and some practice questions. Practising these polynomial division questions will help you score good marks in the examinations.

What is the polynomial?

A polynomial is an algebraic expression of the form a_nxⁿ + a_n−1xⁿ⁻¹ +…………+ a₂x² + a₁x + a₀.

Here, n is either 0 or positive.

a_n, a_n−1,….., a₁, a₀ are coefficients and are real numbers.

Degree: The highest power of variable

Roots: Suppose p(x) is a polynomial and x = k is called the root of p(x) if p(k) = 0.

Unlike numbers, we can perform arithmetic operations on polynomials. Thus, it is possible to divide polynomials by a monomial, binomial or another polynomial.

To perform the polynomial division, it is necessary that the degree of the dividend must be greater than the degree of the divisor.

Polynomial Division Questions and Answers

1. Divide the polynomial 6x³ + 150x² + 5x by 15x.

Solution:

From the given,

Dividend = 6x³ + 150x² + 5x

Divisor = 15x

Here, the degree of dividend, i.e. 3, is greater than the degree of the divisor, i.e. 1.

So, (6x³ + 150x² + 5x)/ 15x

Now divide each term of the numerator by 15 x.

(6x³/15x) + (150x²/15x) + (5x/15x)

= (2/3)x² + 10x + (1/3)

2. Perform the polynomial division by factorisation.

(3x² − 4x + 1) ÷ (x − 1)

Solution:

Given,

(3x² − 4x + 1) ÷ (x − 1)

Degree of the dividend > Degree of the divisor

Consider the numerator:

3x² – 4x + 1

= 3x² – 3x – x + 1

= 3x(x – 1) – 1(x – 1)

= (3x – 1)(x – 1)

Now, (3x² − 4x + 1) / (x − 1) = (3x – 1)(x – 1)/ (x – 1)

= 3x – 1

Therefore, (3x² − 4x + 1) ÷ (x − 1) = 3x – 1

3. Find the quotient and remainder when (x³ – 8) is divided by (x – 2).

Solution:

Given,

Dividend = x³ – 8

Divisor = x – 2

Using the algebraic identity a³ – b³ = (a – b)(a² + ab + b²), we have;

x³ – 8 = x³ – 2³

= (x – 2)(x² + 2x + 4)

Now, (x³ – 8)/(x – 2)

= [(x – 2)(x² + 2x + 4)]/ (x – 2)

= x² + 2x + 4

Thus, remainder = 0 and quotient = x² + 2x + 4.

4. Use the long polynomial division to simplify the following quotient.

(27x³ + 9x² − 3x − 9)/ (3x − 2)

Solution:

Given,

Dividend = 27x³ + 9x² − 3x − 9

Divisor = 3x – 2

By the long division of polynomials, we have;

Therefore, quotient = 9x² + 9x + 5 and remainder = 1.

5. Use polynomial long division to perform the division and then write the dividend in the form p(x) = g(x) q(x) + r(x).

(2x³ + 7x² − 13x − 3) ÷ (2x − 3)

Solution:

Given,

p(x) = 2x³ + 7x² − 13x − 3

g(x) = 2x – 3

Here, q(x) = x² + 5x + 1

r(x) = 0

So, 2x³ + 7x² − 13x − 3 = (2x – 3)(x² + 5x + 1) + 0

6. Using the Synthetic division method, find the quotient and remainder when 7x⁴ − 10x³ + 3x² + 3x − 3 is divided by x − 1.

Solution:

Dividend = 7x⁴ − 10x³ + 3x² + 3x − 3

Divisor = x – 1 (a linear factor)

Coefficients of the numerator, i.e. dividend are:

7 -10 3 3 -3

Let us write the given expression in the Synthetic division format and carry down the leading coefficient as shown below:

Multiply the carry-down value by the zero of the denominator and carry the result up into the next column and then perform the addition.

Continue the above process till we get the last coefficient value.

Here, the last carry-down value is taken as the remainder.

Hence, quotient = 7x³ – 3x² + 3 and remainder = 0.

7. Find the quotient and remainder for (10x² + 53x − 37) ÷ (10x − 7).

Solution:

(10x² + 53x − 37) ÷ (10x − 7)

Here, the numerator can’t be expressed as the factor product of the denominator.

So, let’s perform the long division of polynomials.

Thus, quotient = x + 6 and remainder = 5

8. Divide p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 by g(x) = x² + 3x + 1.

Solution:

Given,

p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2

g(x) = x² + 3x + 1

Therefore, r(x) = 0 and q(x) = 3x² – 4x + 2.

9. If the polynomial p(x) = 3x⁴ – 9x³ + x² + 15x + p completely divisible by 3x² – 5, then find the value of p.

Solution:

Given,

p(x) = 3x⁴ – 9x³ + x² + 15x + p

g(x) = 3x² – 5

Now, by dividing p(x) by g(x), we get;

Given that f(x) is completely divisible by 3x² – 5.

That means the remainder is 0.

p + 10 = 0

p = -10

10. On dividing p(x) = x³ – 3x² + x + 2 by a polynomial g(x), the quotient and remainder were x – 2 and –2x + 4, respectively. Find g(x).

Solution:

Given,

Dividend, p(x) = x³ – 3x² + x + 2

Quotient = x – 2

Remainder = –2x + 4

By Division algorithm of polynomials, we have;

Dividend = Divisor × Quotient + Remainder

Thus, x³ – 3x² + x + 2 = g(x) × (x – 2) + (-2x + 4)

x³ – 3x² + x + 2 – (-2x + 4) = g(x) × (x – 2)

Therefore, g(x) × (x – 2) = x³ – 3x² + 3x – 2

Now, divide (x³ – 3x² + 3x – 2) by (x – 2) to get g(x).

Hence, g(x) = x² – x + 1.

Practice Problems on Polynomial Division

1. 1. Write the quotient and remainder for the following.

      (12a³b⁴ + 12a²b³ + 3ab²)/ 6a²b²

   2. Perform the division (2p² − 9p − 5) ÷ (p − 5).
   3. Using Synthetic division, find the remainder for the following division.

(x³ + 13x² + 42x + 54) ÷ (x + 9)

1. Find the quotient and remainder when p(x) = 4x⁴ + 2x³ − 4x² + 2x + 2 is divided by g(x) = (2x + 1).
2. Verify the division algorithm of polynomials for p(x) = x³ – 3x² + 5x – 3 and g(x) = x² – 2.